Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 2

A process $(X_t{)}_{t\in \mathbb{Z}}$ is weakly stationary when it has finite second moments, a time-independent mean $\mathbb{E}{X}_t=\mu$, and an autocovariance functionthat depends only on the lag.

At lag $h$, the plot shows the sample autocorrelation functionUnder a white noise process, each fixed nonzero-lag sample autocorrelation is approximately $N(0,1/n)$, so the dashed pointwise $95\%$ reference lines are approximately $\pm 1.96/\sqrt{n}$.

The first nonzero-lag bar is well above the upper line, which contradicts the zero autocorrelation expected from white noise. Since the plot then largely cuts off, an moving-average process of order one is a plausible model; with the sampling interval as the time unit this is an $\mathrm{ARMA}(0,1)$ model.

The selected zero-mean autoregressive moving-average process isThe reported maximum-likelihood estimates areUsing the displayed asymptotic standard error gives the Wald confidence intervalThis normal interval is unreliable and likely too narrow because the series has only about twenty observations, the moving-average estimate is near the noninvertibility boundary $\theta =1$, and the same data were used to select the model. The finite-sample likelihood is consequently skewed and model-selection uncertainty is omitted.

The autoregressive process of order one is causal exactly whenbecause then $X_t={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{t-j}$ converges in mean square. Its autocovariance is

Under the intended assumption that the two white-noise sequences are mutually uncorrelated at every pair of times, $X$ and $W$ are uncorrelated. Their sum is therefore weakly stationary with

Strictly, the printed condition $\mathbb{E}[{\epsilon }_t{W}_t]=0$ only at equal times is insufficient. For example, $W_t=(-1{)}^t{\epsilon }_{t-1}$ is itself white noise and is contemporaneously uncorrelated with ${\epsilon }_t$, but the cross-covariance contribution can depend on $t$. The displayed answer therefore uses the standard intended cross-series white-noise assumption $\mathbb{E}[{\epsilon }_t{W}_s]=0$ for all $s,t$.

Using $X_t-\varphi X_{t-1}={\epsilon }_t$ givesUnder the cross-series uncorrelatedness used in part ii, terms in $U_t$ and $U_{t-h}$ involve disjoint white-noise times whenever $|h|>1$. HenceFor completeness,

Part iii and the stated characterization imply that $U$ has a moving-average process of order one representation $U_t={\eta }_t+\theta {\eta }_{t-1}$. Sincewhere $B$ is the backshift operator,Thus $Y$ is a causal autoregressive moving-average process of order $(1,1)$.